False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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FALSE: C prime(1/6) means every relator has length at most six

Statement

If a presentation satisfies $C^{'} (1 / 6)$ , then every relator has length at most $6$ .

Facts & Assumptions

Given: The one-relator presentation $⟨ x_{1}, \dots, x_{7} ∣ x_{1} x_{2} x_{3} x_{4} x_{5} x_{6} x_{7} ⟩$ .

[L1]

$C^{'} (1 / 6)$ bounds the length of pieces as a fraction of the relator length, not the relator length itself (The small-cancellation conditions C(lambda) and C prime(lambda)).

Refutation

technique · direct

1.1

In the displayed one-relator presentation, the symmetrised relator set has no nontrivial piece: distinct cyclic conjugates begin with different letters, and the inverse cyclic conjugates do as well. Hence the $C^{'} (1 / 6)$ condition holds vacuously by [L1].

L1given

2.1

The unique defining relator has length $7$ , which is strictly greater than $6$ .

step 1.1algebra

3.1

So a $C^{'} (1 / 6)$ presentation can have relators longer than $6$ . The statement is false.

step 1.1step 2.1∎

Depends on

The small-cancellation conditions C(lambda) and C prime(lambda)

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

GAP SmallCancellation manual, Chapter 1: Small Cancellation Theory — the classical conditions (standard reference, not scraped)
Jay Williams, Universal Countable Borel Quasi-Orders (standard reference, not scraped)
Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, Section 3.5 (standard reference, not scraped)
Clara Löh, Geometric Group Theory: An Introduction, Section 7.4.1 (standard reference, not scraped)